# Derivative of a Composite Function

How to find the derivative of a composite function g(f(x)): formula (chain rule), 1 example, and its solution.

## FormulaChain Rule

The derivative of a composite function g(f(x)) is

g'(f(x))⋅f'(x).

First differentiate the outer function g.

Then differentiate the inner function f.

## Exampley = (2x^{2} - 1)^{8}, y' = ?

Solution

Solution (Detail)

See the given function.

(2x^{2} - 1) is in (2x^{2} - 1)^{8}.

So this is a composite function.

So use the chain rule.

Write y' =.

See (2x^{2} - 1) as a whole.

And write the derivative of (2x^{2} - 1)^{8}:

8(2x^{2} - 1)^{7}.

Derivative of a Polynomial

Write the derivative of the inner function (2x^{2} - 1):

2⋅2x^{1} - 0.

So y' = 8(2x^{2} - 1)^{7}⋅(2⋅2x^{1} - 0).

(2⋅2x^{1} - 0) = 4x

8⋅4x = 32x

So

y' = 32x(2x^{2} - 1)^{7}

is the derivative of the given function.